A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?

$$\begin{array}{|c|c|}\hline T_1&\underline{\phantom{00}}\,\mathrm m\\T_2&11\,\mathrm m\\T_3&\underline{\phantom{00}}\,\mathrm m\\T_4&\underline{\phantom{00}}\,\mathrm m\\T_5&\underline{\phantom{00}}\,\mathrm m\\\hline\overline T&\underline{\phantom{00}}.2\,\mathrm m\\\hline\end{array}$$

Answer: B

Concepts:caseworkmean

Difficulty rating: 1370

AMC 8AMC 10AMC 12AIME

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Written solution:

Tree $2$ is $11$ meters tall. Since all heights are integers and neighboring trees differ by a factor of $2,$ trees $1$ and $3$ must both be $22$ meters tall.

The first three trees total $22+11+22=55$ meters. The possible integer pairs for trees $4$ and $5$ are $(11,22),$ $(44,22),$ and $(44,88).$

These give averages $17.6,$ $24.2,$ and $37.4,$ respectively. The notebook shows that the average ends in $.2,$ so it must be $24.2.$

Thus, the correct answer is B.